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Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial

Authors Adir Morgan, Shay Solomon, Nicole Wein

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Author Details

Adir Morgan
  • Tel Aviv University, Israel
Shay Solomon
  • Tel Aviv University, Israel
Nicole Wein
  • MIT, Cambridge, MA, USA

Funding

  • Solomon, Shay: Partially supported by the Israel Science Foundation grant No.1991/19.
  • Wein, Nicole: Supported by NSF Grant CCF-1514339.

Acknowledgements

We thank Yosi Hezi and Quanquan Liu for fruitful discussions. We would also like to thank Neal E. Young and Kent Quanrud for correspondence about their prior work.

Cite AsGet BibTex

Adir Morgan, Shay Solomon, and Nicole Wein. Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.DISC.2021.33

Abstract

We revisit the minimum dominating set problem on graphs with arboricity bounded by α. In the (standard) centralized setting, Bansal and Umboh [Bansal and Umboh, 2017] gave an O(α)-approximation LP rounding algorithm, which also translates into a near-linear time algorithm using general-purpose approximation results for explicit mixed packing and covering or pure covering LPs [Koufogiannakis and Young, 2014; Young, 2014; Allen-Zhu and Orecchia, 2019; Quanrud, 2020]. Moreover, [Bansal and Umboh, 2017] showed that it is NP-hard to achieve an asymptotic improvement for the approximation factor. On the other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer [Christoph Lenzen and Roger Wattenhofer, 2010], and Jones et al. [Jones et al., 2013], achieve an approximation factor of O(α²) in linear time. There is a similar situation in the distributed setting: While there is an O(log² n)-round LP-based O(α)-approximation algorithm implied in [Kuhn et al., 2006], the best non-LP-based algorithm by Lenzen and Wattenhofer [Christoph Lenzen and Roger Wattenhofer, 2010] is an implementation of their centralized algorithm, providing an O(α²)-approximation within O(log n) rounds. We address the questions of whether one can achieve an O(α)-approximation algorithm that is elementary, i.e., not based on any LP-based methods, either in the centralized setting or in the distributed setting. We resolve both questions in the affirmative, and en route achieve algorithms that are faster than the state-of-the-art LP-based algorithms. Our contribution is two-fold: 1) In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an O(α)-approximation in linear time. The previous state-of-the-art O(α)-approximation algorithms are (1) LP-based, (2) more complicated, and (3) have super-linear running time. 2) Based on our centralized algorithm, we design a distributed combinatorial O(α)-approximation algorithm in the CONGEST model that runs in O(αlog n) rounds with high probability. Not only does this result provide the first nontrivial non-LP-based distributed o(α²)-approximation algorithm for this problem, it also outperforms the best LP-based distributed algorithm for a wide range of parameters.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph Algorithms
  • Dominating Set
  • Bounded Arboricity
  • Linear time algorithms

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References

  1. Zeyuan Allen-Zhu and Lorenzo Orecchia. Nearly linear-time packing and covering lp solvers. Mathematical Programming, 175(1):307-353, 2019. Google Scholar
  2. Saeed Akhoondian Amiri. Deterministic congest algorithm for mds on bounded arboricity graphs. arXiv preprint, 2021. URL: http://arxiv.org/abs/2102.08076.
  3. Saeed Akhoondian Amiri, Stefan Schmid, and Sebastian Siebertz. Distributed dominating set approximations beyond planar graphs. ACM Transactions on Algorithms (TALG), 15(3):1-18, 2019. Google Scholar
  4. Srinivasa R Arikati, Anil Maheshwari, and Christos D Zaroliagis. Efficient computation of implicit representations of sparse graphs. Discrete Applied Mathematics, 78(1-3):1-16, 1997. Google Scholar
  5. Brenda S Baker. Approximation algorithms for np-complete problems on planar graphs. Journal of the ACM (JACM), 41(1):153-180, 1994. Google Scholar
  6. Nikhil Bansal and Seeun William Umboh. Tight approximation bounds for dominating set on graphs of bounded arboricity. Information Processing Letters, 122:21-24, 2017. Google Scholar
  7. Leonid Barenboim and Michael Elkin. Sublogarithmic distributed mis algorithm for sparse graphs using nash-williams decomposition. Distributed Computing, 22(5-6):363-379, 2010. Google Scholar
  8. Suman K Bera, Amit Chakrabarti, and Prantar Ghosh. Graph coloring via degeneracy in streaming and other space-conscious models. arXiv preprint, 2019. URL: http://arxiv.org/abs/1905.00566.
  9. Suman K. Bera, Noujan Pashanasangi, and C. Seshadhri. Linear time subgraph counting, graph degeneracy, and the chasm at size six. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 38:1-38:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.38.
  10. Suman K. Bera and C. Seshadhri. How the degeneracy helps for triangle counting in graph streams. In Dan Suciu, Yufei Tao, and Zhewei Wei, editors, Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2020, Portland, OR, USA, June 14-19, 2020, pages 457-467. ACM, 2020. URL: https://doi.org/10.1145/3375395.3387665.
  11. Gerth Stølting Brodal and Rolf Fagerberg. Dynamic representation of sparse graphs. In Frank K. H. A. Dehne, Arvind Gupta, Jörg-Rüdiger Sack, and Roberto Tamassia, editors, Algorithms and Data Structures, 6th International Workshop, WADS '99, Vancouver, British Columbia, Canada, August 11-14, 1999, Proceedings, volume 1663 of Lecture Notes in Computer Science, pages 342-351. Springer, 1999. URL: https://doi.org/10.1007/3-540-48447-7_34.
  12. Bernard Chazelle. A minimum spanning tree algorithm with inverse-ackermann type complexity. Journal of the ACM (JACM), 47(6):1028-1047, 2000. Google Scholar
  13. Norishige Chiba and Takao Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on computing, 14(1):210-223, 1985. Google Scholar
  14. Miroslav Chlebík and Janka Chlebíková. Approximation hardness of dominating set problems in bounded degree graphs. Information and Computation, 206(11):1264-1275, 2008. Google Scholar
  15. Andrzej Czygrinow, Michał Hańćkowiak, and Edyta Szymańska. Fast distributed approximation algorithm for the maximum matching problem in bounded arboricity graphs. In International Symposium on Algorithms and Computation, pages 668-678. Springer, 2009. Google Scholar
  16. Andrzej Czygrinow, Michal Hańćkowiak, and Wojciech Wawrzyniak. Fast distributed approximations in planar graphs. In International Symposium on Distributed Computing, pages 78-92. Springer, 2008. Google Scholar
  17. Janosch Deurer, Fabian Kuhn, and Yannic Maus. Deterministic distributed dominating set approximation in the congest model. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, pages 94-103, 2019. Google Scholar
  18. Irit Dinur, Venkatesan Guruswami, Subhash Khot, and Oded Regev. A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput., 34(5):1129-1146, 2005. URL: https://doi.org/10.1137/S0097539704443057.
  19. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 624-633, 2014. Google Scholar
  20. Talya Eden, Reut Levi, and Dana Ron. Testing bounded arboricity. In Artur Czumaj, editor, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 2081-2092. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.136.
  21. Talya Eden, Dana Ron, and Will Rosenbaum. The arboricity captures the complexity of sampling edges. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 52:1-52:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.52.
  22. Talya Eden, Dana Ron, and C. Seshadhri. Faster sublinear approximation of the number of k-cliques in low-arboricity graphs. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1467-1478. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.89.
  23. David Eppstein. Arboricity and bipartite subgraph listing algorithms. Information processing letters, 51(4):207-211, 1994. Google Scholar
  24. Guy Even, Mohsen Ghaffari, and Moti Medina. Distributed set cover approximation: Primal-dual with optimal locality. In 32nd International Symposium on Distributed Computing (DISC 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  25. Michael Fredman and Michael Saks. The cell probe complexity of dynamic data structures. In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 345-354, 1989. Google Scholar
  26. Michael L Fredman and Dan E Willard. Trans-dichotomous algorithms for minimum spanning trees and shortest paths. In Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science, pages 719-725. IEEE, 1990. Google Scholar
  27. Michael R Garey. A guide to the theory of np-completeness. Computers and intractability, 1979. Google Scholar
  28. Mohsen Ghaffari, Christoph Grunau, and Václav Rozhoň. Improved deterministic network decomposition. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2904-2923. SIAM, 2021. Google Scholar
  29. Mohsen Ghaffari and Fabian Kuhn. Derandomizing distributed algorithms with small messages: Spanners and dominating set. In 32nd International Symposium on Distributed Computing (DISC 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  30. Mohsen Ghaffari, Fabian Kuhn, and Yannic Maus. On the complexity of local distributed graph problems. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 784-797, 2017. Google Scholar
  31. Mohsen Ghaffari and Hsin-Hao Su. Distributed degree splitting, edge coloring, and orientations. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2505-2523. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.166.
  32. Gaurav Goel and Jens Gustedt. Bounded arboricity to determine the local structure of sparse graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 159-167. Springer, 2006. Google Scholar
  33. Meng He, Ganggui Tang, and Norbert Zeh. Orienting dynamic graphs, with applications to maximal matchings and adjacency queries. In Hee-Kap Ahn and Chan-Su Shin, editors, Algorithms and Computation - 25th International Symposium, ISAAC 2014, Jeonju, Korea, December 15-17, 2014, Proceedings, volume 8889 of Lecture Notes in Computer Science, pages 128-140. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-13075-0_11.
  34. Lujun Jia, Rajmohan Rajaraman, and Torsten Suel. An efficient distributed algorithm for constructing small dominating sets. Distributed Computing, 15(4):193-205, 2002. Google Scholar
  35. David S Johnson. Approximation algorithms for combinatorial problems. Journal of computer and system sciences, 9(3):256-278, 1974. Google Scholar
  36. Mark Jones, Daniel Lokshtanov, MS Ramanujan, Saket Saurabh, and Ondřej Suchy. Parameterized complexity of directed steiner tree on sparse graphs. In European Symposium on Algorithms, pages 671-682. Springer, 2013. Google Scholar
  37. Haim Kaplan and Shay Solomon. Dynamic representations of sparse distributed networks: A locality-sensitive approach. ACM Transactions on Parallel Computing (TOPC), 8(1):1-26, 2021. Google Scholar
  38. David R Karger, Philip N Klein, and Robert E Tarjan. A randomized linear-time algorithm to find minimum spanning trees. Journal of the ACM (JACM), 42(2):321-328, 1995. Google Scholar
  39. Christos Koufogiannakis and Neal E Young. A nearly linear-time ptas for explicit fractional packing and covering linear programs. Algorithmica, 70(4):648-674, 2014. Google Scholar
  40. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. The price of being near-sighted. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 980-989. ACM Press, 2006. URL: https://doi.org/10.5555/1109557.1109666.
  41. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. Local computation: Lower and upper bounds. Journal of the ACM (JACM), 63(2):1-44, 2016. Google Scholar
  42. Fabian Kuhn and Roger Wattenhofer. Constant-time distributed dominating set approximation. Distributed Computing, 17(4):303-310, 2005. Google Scholar
  43. Christoph Lenzen and Roger Wattenhofer. Minimum dominating set approximation in graphs of bounded arboricity. In Nancy A. Lynch and Alexander A. Shvartsman, editors, Distributed Computing, 24th International Symposium, DISC 2010, Cambridge, MA, USA, September 13-15, 2010. Proceedings, volume 6343 of Lecture Notes in Computer Science, pages 510-524. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-15763-9_48.
  44. Andrew McGregor and Sofya Vorotnikova. A simple, space-efficient, streaming algorithm for matchings in low arboricity graphs. In Raimund Seidel, editor, 1st Symposium on Simplicity in Algorithms, SOSA 2018, January 7-10, 2018, New Orleans, LA, USA, volume 61 of OASICS, pages 14:1-14:4. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/OASIcs.SOSA.2018.14.
  45. Adir Morgan, Shay Solomon, and Nicole Wein. Algorithms for the minimum dominating set problem in bounded arboricity graphs: Simpler, faster, and combinatorial. arXiv preprint, 2021. URL: http://arxiv.org/abs/2102.10077.
  46. Jose C Nacher and Tatsuya Akutsu. Minimum dominating set-based methods for analyzing biological networks. Methods, 102:57-63, 2016. Google Scholar
  47. Krzysztof Onak, Baruch Schieber, Shay Solomon, and Nicole Wein. Fully dynamic MIS in uniformly sparse graphs. In Ioannis Chatzigiannakis, Christos Kaklamanis, Dániel Marx, and Donald Sannella, editors, 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, volume 107 of LIPIcs, pages 92:1-92:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.92.
  48. Merav Parter, David Peleg, and Shay Solomon. Local-on-average distributed tasks. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 220-239. SIAM, 2016. Google Scholar
  49. David Peleg and Shay Solomon. Dynamic (1+ε)-approximate matchings: A density-sensitive approach. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 712-729. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch51.
  50. Kent Quanrud. Nearly linear time approximations for mixed packing and covering problems without data structures or randomization. In Symposium on Simplicity in Algorithms, pages 69-80. SIAM, 2020. Google Scholar
  51. Václav Rozhoň and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 350-363, 2020. Google Scholar
  52. Chao Shen and Tao Li. Multi-document summarization via the minimum dominating set. In Proceedings of the 23rd International Conference on Computational Linguistics (Coling 2010), pages 984-992, 2010. Google Scholar
  53. Shay Solomon and Nicole Wein. Improved dynamic graph coloring. In 26th Annual European Symposium on Algorithms (ESA 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  54. Hsin-Hao Su and Hoa T. Vu. Distributed dense subgraph detection and low outdegree orientation. In Hagit Attiya, editor, 34th International Symposium on Distributed Computing, DISC 2020, October 12-16, 2020, Virtual Conference, volume 179 of LIPIcs, pages 15:1-15:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.15.
  55. Robert Endre Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM (JACM), 22(2):215-225, 1975. Google Scholar
  56. Peng-Jun Wan, Khaled M Alzoubi, and Ophir Frieder. Distributed construction of connected dominating set in wireless ad hoc networks. In Proceedings. Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies, volume 3, pages 1597-1604. IEEE, 2002. Google Scholar
  57. Neal E Young. Nearly linear-work algorithms for mixed packing/covering and facility-location linear programs. arXiv preprint, 2014. URL: http://arxiv.org/abs/1407.3015.
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